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LongLine[1] - The Long Line Richard Koch 1 Introduction...

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The Long Line Richard Koch November 24, 2005 1 Introduction Before this class began, I asked several topologists what I should cover in the first term. Everyone told me the same thing: go as far as the classification of compact surfaces. Having done my duty, I feel free to talk about more general results and give details about one of my favorite examples. We classified all compact connected 2-dimensional manifolds. You may wonder about the corresponding classification in other dimensions. It is fairly easy to prove that the only compact connected 1-dimensional manifold is the circle S 1 ; the book sketches a proof of this and I have nothing to add. In dimension greater than or equal to four, it has been proved that a complete classification is impossible (although there are many interesting theorems about such manifolds). The idea of the proof is interesting: for each finite presentation of a group by generators and re- lations, one can construct a compact connected 4-manifold with that group as fundamental group. Logicians have proved that the word problem for finitely presented groups cannot be solved. That is, if I describe a group G by giving a finite number of generators and a finite number of relations, and I describe a second group H similarly, it is not possible to find an algorithm which will determine in all cases whether G and H are isomorphic. A complete classification of 4-manifolds, however, would give such an algorithm. As for the theory of compact connected three dimensional manifolds, this is a very ex- citing time to be alive if you are interested in that theory. Three dimensional compact manifolds have been studied intensively since the work of Poincare around 1900. In the 1970’s, Bill Thurston expanded this theory dramatically; roughly speaking, he showed that every such manifold can be canonically cut into pieces of a special nature, and then conjectured that each of these pieces can be given a particular kind of geometry. The geometry dramatically decreases the possible topologies of that piece. See for ex- ample, http://mathworld.wolfram.com/ThurstonsGeometrizationConjecture.html for more details. 1
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Around 1980, a possible approach to proving the Thurston conjecture was developed by R. S. Hamilton. Two faculty members at Oregon, Peng Lu and Jim Isenberg, are experts in this theory. In 2002, G. Perelman announced a proof of the full Thurston conjectures using Hamilton’s technique. Experts are verifying his proof now; the experts believe that there is a good chance it is correct. Perelman’s proof will not complete the classification, but it will push us astonishingly closer to that goal. One special case of Perelman’s result would give all possible compact 3-manifolds with finite fundamental group. In particular, it would show that S 3 is the only compact 3- manifold with trivial fundamental group, so that every closed loop can be deformed to a point. Proving this special case is one of seven unsolved problems for which the Clay Institute offers a prize of $1,000,000 for a solution.
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